In electronics a NOT gate is more commonly called an inverter. The circle on the symbol is called a bubble, and is used in logic diagrams to indicate a logic negation between the external logic state and the internal logic state (1 to 0 or vice versa).
A weather watch, on the other hand, typically refers to an alert issued to indicate that conditions are favorable for the development of dangerous weather patterns, although the dangerous weather conditions themselves are not currently present.
Severe Thunderstorm Watch This is issued by the National Weather Service when conditions are favorable for the development of severe thunderstorms in and close to the watch area. A severe thunderstorm by definition is a thunderstorm that produces one inch hail or larger in diameter and/or winds equal or exceed 58 miles an hour. The size of the watch can vary depending on the weather situation. They are usually issued for a duration of 4 to 8 hours.